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Theorem eengtrkg 25865

Description: The geometry structure for 𝔼↑𝑁 is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

**Assertion**
Ref	Expression
eengtrkg	⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Proof of Theorem eengtrkg Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑖 𝑝 𝑠 𝑡 𝑢 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6203	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ V)
2		simpl 473	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
3		simprl 794	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
4		eengbas 25861	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
5	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
6	3, 5	eleqtrrd 2704	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
7		simprr 796	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
8	7, 5	eleqtrrd 2704	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
9		axcgrrflx 25794	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
10	2, 6, 8, 9	syl3anc 1326	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩)
11		eqid 2622	. . . . . . . . 9 ⊢ (Base‘(EEG‘𝑁)) = (Base‘(EEG‘𝑁))
12		eqid 2622	. . . . . . . . 9 ⊢ (dist‘(EEG‘𝑁)) = (dist‘(EEG‘𝑁))
13	2, 11, 12, 3, 7, 7, 3	ecgrtg 25863	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑦, 𝑥⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥)))
14	10, 13	mpbid 222	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
15	14	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥))
16		simpl 473	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
17		simpr1 1067	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
18		simpr2 1068	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
19		simpr3 1069	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
20	16, 11, 12, 17, 18, 19, 19	ecgrtg 25863	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧)))
21	6	3adantr3 1222	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
22	8	3adantr3 1222	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
23	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
24	19, 23	eleqtrrd 2704	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
25		axcgrid 25796	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
26	16, 21, 22, 24, 25	syl13anc 1328	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑧, 𝑧⟩ → 𝑥 = 𝑦))
27	20, 26	sylbird 250	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑧 ∈ (Base‘(EEG‘𝑁)))) → ((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))
28	27	ralrimivvva 2972	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))
29	1, 15, 28	jca32 558	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))))
30		eqid 2622	. . . . . 6 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
31	11, 12, 30	istrkgc 25353	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_C ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑦(dist‘(EEG‘𝑁))𝑥) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑧(dist‘(EEG‘𝑁))𝑧) → 𝑥 = 𝑦))))
32	29, 31	sylibr 224	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_C)
33	2, 11, 30, 3, 3, 7	ebtwntg 25862	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)))
34		axbtwnid 25819	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
35	2, 8, 6, 34	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑥⟩ → 𝑦 = 𝑥))
36	33, 35	sylbird 250	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑦 = 𝑥))
37	36	imp 445	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑦 = 𝑥)
38	37	eqcomd 2628	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥)) → 𝑥 = 𝑦)
39	38	ex 450	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
40	39	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦))
41		simpll 790	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
42	6	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
43	8	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
44	3	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
45	7	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
46		simpr1 1067	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
47	41, 44, 45, 46, 24	syl13anc 1328	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
48		simpr2 1068	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
49	41, 4	syl 17	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
50	48, 49	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (𝔼‘𝑁))
51		simpr3 1069	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
52	51, 49	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (𝔼‘𝑁))
53		axpasch 25821	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁) ∧ 𝑧 ∈ (𝔼‘𝑁)) ∧ (𝑢 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) → ∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩)))
54	41, 42, 43, 47, 50, 52, 53	syl132anc 1344	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) → ∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩)))
55	41, 11, 30, 44, 46, 48	ebtwntg 25862	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑢 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
56	41, 11, 30, 45, 46, 51	ebtwntg 25862	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑣 Btwn ⟨𝑦, 𝑧⟩ ↔ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)))
57	55, 56	anbi12d 747	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑣 Btwn ⟨𝑦, 𝑧⟩) ↔ (𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧))))
58		simplll 798	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
59	48	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
60	45	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
61		simpr 477	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (𝔼‘𝑁))
62	49	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
63	61, 62	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
64	58, 11, 30, 59, 60, 63	ebtwntg 25862	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝑎 Btwn ⟨𝑢, 𝑦⟩ ↔ 𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦)))
65	51	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
66	44	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
67	58, 11, 30, 65, 66, 63	ebtwntg 25862	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (𝑎 Btwn ⟨𝑣, 𝑥⟩ ↔ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥)))
68	64, 67	anbi12d 747	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → ((𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩) ↔ (𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
69	49, 68	rexeqbidva 3155	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)(𝑎 Btwn ⟨𝑢, 𝑦⟩ ∧ 𝑎 Btwn ⟨𝑣, 𝑥⟩) ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
70	54, 57, 69	3imtr3d 282	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
71	70	ralrimivvva 2972	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
72	71	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))))
73		simpl 473	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
74		elpwi 4168	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
75	74	ad2antrl 764	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
76	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
77	75, 76	sseqtr4d 3642	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑠 ⊆ (𝔼‘𝑁))
78		elpwi 4168	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
79	78	ad2antll 765	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
80	79, 76	sseqtr4d 3642	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → 𝑡 ⊆ (𝔼‘𝑁))
81		simpll 790	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑁 ∈ ℕ)
82		simplrl 800	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑠 ⊆ (𝔼‘𝑁))
83		simplrr 801	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → 𝑡 ⊆ (𝔼‘𝑁))
84		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)
85		axcont 25856	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩)) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
86	81, 82, 83, 84, 85	syl13anc 1328	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) ∧ ∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩) → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩)
87	86	ex 450	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ⊆ (𝔼‘𝑁) ∧ 𝑡 ⊆ (𝔼‘𝑁))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
88	73, 77, 80, 87	syl12anc 1324	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ → ∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩))
89		simplll 798	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑁 ∈ ℕ)
90		simplr 792	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑎 ∈ (𝔼‘𝑁))
91	76	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
92	90, 91	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
93	79	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
94		simprr 796	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
95	93, 94	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
96	75	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
97		simprl 794	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑠)
98	96, 97	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
99	89, 11, 30, 92, 95, 98	ebtwntg 25862	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
100	99	2ralbidva 2988	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑎 ∈ (𝔼‘𝑁)) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
101	76, 100	rexeqbidva 3155	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 Btwn ⟨𝑎, 𝑦⟩ ↔ ∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦)))
102		simplll 798	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑁 ∈ ℕ)
103	75	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑠 ⊆ (Base‘(EEG‘𝑁)))
104		simprl 794	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ 𝑠)
105	103, 104	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
106	79	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ⊆ (Base‘(EEG‘𝑁)))
107		simprr 796	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
108	106, 107	sseldd 3604	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
109		simplr 792	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑏 ∈ (𝔼‘𝑁))
110	76	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
111	109, 110	eleqtrd 2703	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
112	102, 11, 30, 105, 108, 111	ebtwntg 25862	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) ∧ (𝑥 ∈ 𝑠 ∧ 𝑦 ∈ 𝑡)) → (𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
113	112	2ralbidva 2988	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) ∧ 𝑏 ∈ (𝔼‘𝑁)) → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
114	76, 113	rexeqbidva 3155	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑏 ∈ (𝔼‘𝑁)∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
115	88, 101, 114	3imtr3d 282	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁)) ∧ 𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁)))) → (∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
116	115	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
117	40, 72, 116	3jca 1242	. . . . 5 ⊢ (𝑁 ∈ ℕ → (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))) ∧ ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦))))
118	11, 12, 30	istrkgb 25354	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_B ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))((𝑢 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑣 ∈ (𝑦(Itv‘(EEG‘𝑁))𝑧)) → ∃𝑎 ∈ (Base‘(EEG‘𝑁))(𝑎 ∈ (𝑢(Itv‘(EEG‘𝑁))𝑦) ∧ 𝑎 ∈ (𝑣(Itv‘(EEG‘𝑁))𝑥))) ∧ ∀𝑠 ∈ 𝒫 (Base‘(EEG‘𝑁))∀𝑡 ∈ 𝒫 (Base‘(EEG‘𝑁))(∃𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑦) → ∃𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))))
119	1, 117, 118	sylanbrc 698	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_B)
120	32, 119	elind 3798	. . 3 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiG_C ∩ TarskiG_B))
121		simplll 798	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
122	3	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
123	121, 4	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
124	122, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
125	7	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
126	125, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
127		simplr1 1103	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
128	127, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑧 ∈ (𝔼‘𝑁))
129		simplr2 1104	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (Base‘(EEG‘𝑁)))
130	129, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑢 ∈ (𝔼‘𝑁))
131		simplr3 1105	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
132	131, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (𝔼‘𝑁))
133		simpr1 1067	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
134	133, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (𝔼‘𝑁))
135		simpr2 1068	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑐 ∈ (Base‘(EEG‘𝑁)))
136	135, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑐 ∈ (𝔼‘𝑁))
137		simpr3 1069	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (Base‘(EEG‘𝑁)))
138	137, 123	eleqtrrd 2704	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → 𝑣 ∈ (𝔼‘𝑁))
139		3anass 1042	. . . . . . . . . . . 12 ⊢ (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) ↔ ((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))))
140		ax5seg 25818	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑧 ∈ (𝔼‘𝑁) ∧ 𝑢 ∈ (𝔼‘𝑁) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑏 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
141	139, 140	syl5bir 233	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑧 ∈ (𝔼‘𝑁) ∧ 𝑢 ∈ (𝔼‘𝑁) ∧ 𝑎 ∈ (𝔼‘𝑁)) ∧ (𝑏 ∈ (𝔼‘𝑁) ∧ 𝑐 ∈ (𝔼‘𝑁) ∧ 𝑣 ∈ (𝔼‘𝑁))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
142	121, 124, 126, 128, 130, 132, 134, 136, 138, 141	syl333anc 1358	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) → ⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩))
143	121, 11, 30, 122, 127, 125	ebtwntg 25862	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑦 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
144	121, 11, 30, 131, 135, 133	ebtwntg 25862	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)))
145	143, 144	3anbi23d 1402	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ↔ (𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐))))
146	121, 11, 12, 122, 125, 131, 133	ecgrtg 25863	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
147	121, 11, 12, 125, 127, 133, 135	ecgrtg 25863	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)))
148	146, 147	anbi12d 747	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ↔ ((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐))))
149	121, 11, 12, 122, 129, 131, 137	ecgrtg 25863	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ↔ (𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣)))
150	121, 11, 12, 125, 129, 133, 137	ecgrtg 25863	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))
151	149, 150	anbi12d 747	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → ((⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩) ↔ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣))))
152	148, 151	anbi12d 747	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩)) ↔ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))))
153	145, 152	anbi12d 747	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ 𝑏 Btwn ⟨𝑎, 𝑐⟩) ∧ ((⟨𝑥, 𝑦⟩Cgr⟨𝑎, 𝑏⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑏, 𝑐⟩) ∧ (⟨𝑥, 𝑢⟩Cgr⟨𝑎, 𝑣⟩ ∧ ⟨𝑦, 𝑢⟩Cgr⟨𝑏, 𝑣⟩))) ↔ ((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣))))))
154	121, 11, 12, 127, 129, 135, 137	ecgrtg 25863	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (⟨𝑧, 𝑢⟩Cgr⟨𝑐, 𝑣⟩ ↔ (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
155	142, 153, 154	3imtr3d 282	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑏 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑐 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑣 ∈ (Base‘(EEG‘𝑁)))) → (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
156	155	ralrimivvva 2972	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑧 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑢 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑎 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
157	156	ralrimivvva 2972	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
158	157	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)))
159		simpll 790	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑁 ∈ ℕ)
160	6	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑥 ∈ (𝔼‘𝑁))
161	8	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑦 ∈ (𝔼‘𝑁))
162		simprl 794	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
163	159, 4	syl 17	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
164	162, 163	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑎 ∈ (𝔼‘𝑁))
165		simprr 796	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
166	165, 163	eleqtrrd 2704	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → 𝑏 ∈ (𝔼‘𝑁))
167		axsegcon 25807	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) ∧ (𝑎 ∈ (𝔼‘𝑁) ∧ 𝑏 ∈ (𝔼‘𝑁))) → ∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩))
168	159, 160, 161, 164, 166, 167	syl122anc 1335	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → ∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩))
169		simplll 798	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
170	3	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑥 ∈ (Base‘(EEG‘𝑁)))
171		simpr 477	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑧 ∈ (𝔼‘𝑁))
172	163	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
173	171, 172	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑧 ∈ (Base‘(EEG‘𝑁)))
174	7	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑦 ∈ (Base‘(EEG‘𝑁)))
175	169, 11, 30, 170, 173, 174	ebtwntg 25862	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑧⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧)))
176		simplrl 800	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑎 ∈ (Base‘(EEG‘𝑁)))
177		simplrr 801	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → 𝑏 ∈ (Base‘(EEG‘𝑁)))
178	169, 11, 12, 174, 173, 176, 177	ecgrtg 25863	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → (⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩ ↔ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
179	175, 178	anbi12d 747	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) ∧ 𝑧 ∈ (𝔼‘𝑁)) → ((𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩) ↔ (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏))))
180	163, 179	rexeqbidva 3155	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → (∃𝑧 ∈ (𝔼‘𝑁)(𝑦 Btwn ⟨𝑥, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩Cgr⟨𝑎, 𝑏⟩) ↔ ∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏))))
181	168, 180	mpbid 222	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) ∧ (𝑎 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑏 ∈ (Base‘(EEG‘𝑁)))) → ∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
182	181	ralrimivva 2971	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ (Base‘(EEG‘𝑁)) ∧ 𝑦 ∈ (Base‘(EEG‘𝑁)))) → ∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
183	182	ralrimivva 2971	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))
184	1, 158, 183	jca32 558	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))))
185	11, 12, 30	istrkgcb 25355	. . . . 5 ⊢ ((EEG‘𝑁) ∈ TarskiG_CB ↔ ((EEG‘𝑁) ∈ V ∧ (∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑧 ∈ (Base‘(EEG‘𝑁))∀𝑢 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∀𝑐 ∈ (Base‘(EEG‘𝑁))∀𝑣 ∈ (Base‘(EEG‘𝑁))(((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ 𝑏 ∈ (𝑎(Itv‘(EEG‘𝑁))𝑐)) ∧ (((𝑥(dist‘(EEG‘𝑁))𝑦) = (𝑎(dist‘(EEG‘𝑁))𝑏) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑏(dist‘(EEG‘𝑁))𝑐)) ∧ ((𝑥(dist‘(EEG‘𝑁))𝑢) = (𝑎(dist‘(EEG‘𝑁))𝑣) ∧ (𝑦(dist‘(EEG‘𝑁))𝑢) = (𝑏(dist‘(EEG‘𝑁))𝑣)))) → (𝑧(dist‘(EEG‘𝑁))𝑢) = (𝑐(dist‘(EEG‘𝑁))𝑣)) ∧ ∀𝑥 ∈ (Base‘(EEG‘𝑁))∀𝑦 ∈ (Base‘(EEG‘𝑁))∀𝑎 ∈ (Base‘(EEG‘𝑁))∀𝑏 ∈ (Base‘(EEG‘𝑁))∃𝑧 ∈ (Base‘(EEG‘𝑁))(𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧) ∧ (𝑦(dist‘(EEG‘𝑁))𝑧) = (𝑎(dist‘(EEG‘𝑁))𝑏)))))
186	184, 185	sylibr 224	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG_CB)
187	11, 30	elntg 25864	. . . . 5 ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ (Base‘(EEG‘𝑁)), 𝑦 ∈ ((Base‘(EEG‘𝑁)) ∖ {𝑥}) ↦ {𝑧 ∈ (Base‘(EEG‘𝑁)) ∣ (𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧))}))
188	11, 12, 30	istrkgl 25357	. . . . 5 ⊢ ((EEG‘𝑁) ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})} ↔ ((EEG‘𝑁) ∈ V ∧ (LineG‘(EEG‘𝑁)) = (𝑥 ∈ (Base‘(EEG‘𝑁)), 𝑦 ∈ ((Base‘(EEG‘𝑁)) ∖ {𝑥}) ↦ {𝑧 ∈ (Base‘(EEG‘𝑁)) ∣ (𝑧 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑧(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑧))})))
189	1, 187, 188	sylanbrc 698	. . . 4 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
190	186, 189	elind 3798	. . 3 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
191	120, 190	elind 3798	. 2 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ ((TarskiG_C ∩ TarskiG_B) ∩ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})))
192		df-trkg 25352	. 2 ⊢ TarskiG = ((TarskiG_C ∩ TarskiG_B) ∩ (TarskiG_CB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
193	191, 192	syl6eleqr 2712	1 ⊢ (𝑁 ∈ ℕ → (EEG‘𝑁) ∈ TarskiG)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 [wsbc 3435 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ⟨cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℕcn 11020 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 TarskiG_Ccstrkgc 25330 TarskiG_Bcstrkgb 25331 TarskiG_CBcstrkgcb 25332 Itvcitv 25335 LineGclng 25336 𝔼cee 25768 Btwn cbtwn 25769 Cgrccgr 25770 EEGceeng 25857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-ds 15964 df-itv 25337 df-lng 25338 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-ee 25771 df-btwn 25772 df-cgr 25773 df-eeng 25858

This theorem is referenced by: (None)

W3C validator